VARIABLE STARS
195
/
Now
wtR _ n 2 R 3 _ 4tt 2 3 3t t p c
~9o ~ QM ~ IP = IPGft pf,
- 12-2ya
(131-3),
by (130-3) introducing the value Pe / Pm = 54-36.
h or the two groups of Cepheids Nos. 5—9 and 10—14 we found
(ya)* = -342, -302.
Substituting in (131-2), we see that at the boundary is between — AP X
and — £P x . Since the pressure cannot become negative P x must not
exceed unity. Hence there is an upper limit to £ x or SP/P between i and £.
Although the observed values of 8P/P in Table 25 do not reach so high
a limit, it seems possible that the vanishing of the pressure is effective
in setting the limit to the amplitude attained in Cepheid pulsation and
that the more typical Cepheids reach this limit. It has been explained
that the values of 8P/P are probably systematically too low since the
spectroscopic determination of SP refers to the integrated light of a hemi
sphere. Allowing for this the amplitude generally attained seems to be
about half the theoretical limit. The discrepancy may well be due to certain
imperfections in the theory. When P x = 1 it is clearly illegitimate to
neglect the squares of the amplitudes as we have done; a correction on
account of this is necessary, but I do not think this is the whole cause of
the difference. It is not sufficient that the total pressure P should remain
positive; both p Q and p R must be positive. In the outermost part of the
star p R and p G become out of phase with one another. This phase-difference
is not shown in our theoretical equations because it is a result of the
failure of the adiabatic approximation in this region; but we supply the
gap by our observational knowledge. The critical time is when the star
is at its greatest expansion. At that time we know from observation that
the star is near its mean luminosity. Since the outflowing stream of
radiation has its mean intensity it seems permissible to assume that p R
has its mean value which for an average Cepheid is about |P 0 . Then since
p G t is not negative we must have P 0 + SP > |P 0 which requires that
the amplitude of P x shall not exceed By (131-2) the corresponding
limit of is to in good agreement with observation. This argument
depends on a patched-up treatment of the non-adiabatic region which
may be fallacious, and it is put forward only as a suggestion*.
* I believe that some years ago, when closely engaged with this branch of the
subject, I came to the conclusion that the whole argument given in this section
was fallacious; but I cannot remember the reasons, and do not now see the flaw, if
any. In Monthly Notices, 79, p. 22 (1918) I stated that the discussion would appear
in Part II of the paper, but for reasons now forgotten withheld it when Part II was
published.
Hence